Validated numerics for the superharmonic instability of a steep Stokes wave

Files and how to check them. stokes-instability.pdf is the article. stokes-instability-supplement.zip holds the certificate, its specification, the two verification programs, eight test inputs, and the three written proofs exactly as reviewed, with their replay scripts and reviews; it also keeps the two superseded versions of the main proof, the records of a section-by-section audit, and earlier floating-point experiments. Running "python3 -I check.py" inside the archive (CPython 3.12, python-flint 0.9.0) repeats the verification in about three minutes. Result. For two-dimensional irrotational gravity waves on infinite depth (g = 1, wavelength 2π) there is a Stokes wave of steepness exactly 69/500 = 0.138 such that the linearized water-wave equations about it admit a 2π-periodic solution growing like e^{λt}, with λ real and 0.19735494607379934 ≤ λ ≤ 0.19735494607379940. Computations have long shown that steep Stokes waves on deep water are unstable to perturbations of their own period, with onset at the first maximum of the wave energy (H/λ ≈ 0.1366); this is a proof for one specific wave. Method. The wave (Babenko's equation) and the eigenpair (the co-periodic spectral problem in the conformal form of Dyachenko and Pelinovsky) are enclosed together by a Newton–Kantorovich estimate in weighted ℓ¹ Fourier spaces, written in the auxiliary conformal variable of Lushnikov, Dyachenko and Silantyev; the eigenpair estimate holds for every wave in the enclosure. Two analytic theorems pass from the conformal equations to the Euler equations and to the Zakharov–Craig–Sulem linearization. The finite list of inequalities is checked with exact rationals and Arb balls (python-flint over FLINT) by two programs written separately from a fixed specification. Assumptions and limitations. The proof assumes that the hardware, CPython 3.12, python-flint 0.9.0 and FLINT 3.6.0 behave as documented. No human specialist has gone through the written proofs line by line so far. Use of AI. The work was directed by the author, who is responsible for its content. The computations, proofs and text were produced with Claude Opus 5.5 (Anthropic) and GPT-6 Astra (OpenAI) between 7 and 9 November 2026. Each written proof was reviewed by the model that did not write it, and the main proof was audited section by section by five further GPT-6 Astra referees. Section 11 of the article gives the details.